Asymptotics for out of sample tests of Granger causality

This paper presents analytical, Monte Carlo and empirical evidence concerning out-of-sample tests of Granger causality. The environment is one in which the relative predictive ability of two nested parametric regression models is of interest. Results are provided for three statistics: a regression-based statistic suggested by Granger and Newbold [1977. Forecasting Economic Time Series. Academic Press Inc., London], a t-type statistic comparable to those suggested by Diebold and Mariano [1995, Comparing Predictive Accuracy. Journal of Business and Economic Statistics, 13, 253–263] and West [1996. Asymptotic Inference About Predictive Ability, Econometrica, 64, 1067–1084], and an F-type statistic akin to Theilʹs U. Since the asymptotic distributions under the null are nonstandard, tables of asymptotically valid critical values are provided. Monte Carlo evidence supports the theoretical results. An empirical example evaluates the predictive content of the Chicago Fed National Activity Index for growth in Industrial Production and core PCE-based inflation.